MATH2051_08

MATH2051_08 - Durham University EXAMINATION PAPER date...

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Unformatted text preview: Durham University EXAMINATION PAPER date May/June 2008 exam code MATH2051/01 description Numerical Analysis II Time allowed: 3 hours Examination material provided: None Instructions: Credit will be given for the best FOUR answers from Section A and the best THREE answers from Section B. Questions in Section B carry TWICE as many marks as those in Section A. Approved electronic calculators may be used. ED01/2008 Durham University Copyright continued 2 MATH2051/01 SECTION A 1. The sequence { p n } generated by the formula p n +1 = 1 4 parenleftbigg 3 p n + 17 p 3 n parenrightbigg converges for suitable positive real values of p . To what limit does it converge and what is the order of convergence? 2. (i) A certain well behaved function f ( x ) has the following values . 40 . 44 . 48 . 52 . 56 . 1987 0 . 2182 0 . 2377 0 . 2571 0 . 2764 Form a divided difference table and compute f (0 . 462) as accurately as you can. (ii) Given that g ( x ) = 1 exp( x 2 ) 2 x sin x evaluate g (0 . 001) correct to three decimal places. 3. (a) Prove that for sufficiently smooth f , the error in the Central Difference formula takes the form f ( x + h ) f ( x h ) 2 h f ( x ) = h 2 6 f (3) ( c ) . (b) Using the approximation f ( x ) f ( x + h ) f ( x h ) 2 h and the data from Question 2(i) compute f (0 . 48) as accurately as you can. 4. Let A := parenleftbigg 1 10 4 10 4 1 parenrightbigg , b := parenleftbigg 10 4 10 4 parenrightbigg . Solve the linear system Ax = b with Gaussian elimination, once with and once without partial pivoting, using 3 digit arithmetic (meaning after each arithmetic operation the intermediate result is rounded to 3 significant figures)....
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MATH2051_08 - Durham University EXAMINATION PAPER date...

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